Finding all tangent angles

\tan(\theta+180)=\tan\theta

This means that, if we know one solution for \theta, we can find another by adding 180\degree to that angle.

Negative tangent angles

\tan(-\theta)=-\tan\theta

The tangent of a negative angle is the negative of the tangent of the positive angle

Solving tangent equations

Solve tan\theta=1 for values of \theta between 0\degree and 540\degree.

\tan^{-1}(1)=45\degree
45\degree+180\degree=225\degree
225\degree+180\degree=405\degree
Answer: \theta=45\degree,225\degree,405\degree

Solve tan\theta=-1 for values of \theta between 0\degree and 540\degree.

\tan^{-1}(-1)=-45\degree
-45\degree+180\degree=135\degree
135\degree+180\degree=315\degree
315\degree+180\degree=495\degree
Answer: \theta=135\degree,315\degree,495\degree

Solve tan\theta=\sqrt{3} for values of \theta between 0\degree and 540\degree.

\tan^{-1}(\sqrt{3})=60\degree
60\degree+180\degree=240\degree
240\degree+180\degree=420\degree
Answer: \theta=60\degree,240\degree,420\degree

flashcards

Question	Answer
What is the periodicity property of \tan\theta?	\tan(\theta+180^\circ) = \tan\theta holds, so adding 180^\circ yields another solution.
What is \tan(-\theta) equal to?	\tan(-\theta) = -\tan\theta; the tangent of a negative angle is the negative of the tangent of the positive angle.
How do you find all tangent angles given one solution?	Add 180^\circ repeatedly to the known solution to generate further angles within the required range.
Solve \tan\theta = 1 for \theta between 0^\circ and 540^\circ.	\theta = 45^\circ, 225^\circ, 405^\circ (from \tan^{-1}(1)=45^\circ, then add 180^\circ twice).
Solve \tan\theta = -1 for \theta between 0^\circ and 540^\circ.	\theta = 135^\circ, 315^\circ, 495^\circ (from \tan^{-1}(-1)=-45^\circ; add 180^\circ three times to stay in range).
Solve \tan\theta = \sqrt{3} for \theta between 0^\circ and 540^\circ.	\theta = 60^\circ, 240^\circ, 420^\circ (from \tan^{-1}(\sqrt{3})=60^\circ, then add 180^\circ twice).

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